Investigating Generic Methods to Solve Hopf Bifurcation Problems in Algebraic Biology

Abstract

Abstract. Symbolic methods for investigating Hopf bifurcation problems of vector fields arising in the context of algebraic biology have recently obtained renewed attention. However, the symbolic investigations have not been fully algorithmic but required a sequence of symbolic computations intervened with ad hoc insights and decisions made by a human. In this paper we discuss the use of generic methods to reduce questions on the existence of Hopf bifurcations in parameterized polynomial vector fields to quantifier elimination problems over the reals combined with simplification techniques available in REDLOG. We can reconstruct most of the results given in the literature within a few seconds of computation time. As no tedious hand computations are involved we presume that the use of these generic methods will be a useful tool for investigating other examples.

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

“…The idea of positive quantifier elimination is to have a variant of the elimination procedure which systematically exploits that assumption [57,64]. First implementations of that idea, as the one used with our life science application in Sect.…”

“…Our example discussed here dates back to 2008. It is to our knowledge the first fully automatic symbolic investigation of Hopf bifurcations for a biologically relevant model [57,64]. 19 Boulier et al have studied a model related to the gene regulatory network controlling the circadian clock of a certain unicellular green alga [3].…”

On Muldowney’s Criteria for Polynomial Vector Fields with Constraints

“…If all variables and parameters are known to be positive, the technique of positive quantifier elimination [14,15] can be used, which was first used to solve semi-algebraic criteria for the existence of Hopf bifurcation fixed points [16,17] arising in the context of chemistry and algebraic biology [18][19][20][21].…”

Algorithmic Global Criteria for Excluding Oscillations

“…Using the theory of Hopf bifurcations some non-numeric algorithmic methods have been recently developed to determine ranges of parameters for which some small stable limit cycle will occur in the system (Sturm et al 2009;Niu and Wang 2008;Sturm and Weber 2008;Boulier et al 2007Boulier et al , 2008Weber 2000, 2002). These algorithms give exact conditions for the existence of fixed points undergoing a Poincaré-Andronov-Hopf bifurcation that give birth to a small stable limit cycle under some general conditions which can be made algorithmic, too.…”

“…3. Using so called positive quantifier elimination (Sturm and Weber 2008;Sturm et al 2009)-a technique requiring that all parameters and variables are restricted to positive values, which is very often the case in the context of computational biology-we provide an in depth study of a family of computational examples in Sect. 4.…”